Theorem cfss 10267

Description: There is a cofinal subset of 𝐴 of cardinality (cf‘𝐴). (Contributed by Mario Carneiro, 24-Jun-2013.)

Hypothesis

Ref	Expression
cfss.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
cfss	⊢ (Lim 𝐴 → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ (cf‘𝐴) ∧ ∪ 𝑥 = 𝐴))

Distinct variable group:   𝑥,𝐴

Proof of Theorem cfss

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cfss.1	. . . . . 6 ⊢ 𝐴 ∈ V
2	1	cflim3 10264	. . . . 5 ⊢ (Lim 𝐴 → (cf‘𝐴) = ∩ 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝑥))
3		fvex 6901	. . . . . . 7 ⊢ (card‘𝑥) ∈ V
4	3	dfiin2 5002	. . . . . 6 ⊢ ∩ 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝑥) = ∩ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥)}
5		cardon 9949	. . . . . . . . . 10 ⊢ (card‘𝑥) ∈ On
6		eleq1 2854	. . . . . . . . . 10 ⊢ (𝑦 = (card‘𝑥) → (𝑦 ∈ On ↔ (card‘𝑥) ∈ On))
7	5, 6	mpbiri 261	. . . . . . . . 9 ⊢ (𝑦 = (card‘𝑥) → 𝑦 ∈ On)
8	7	rexlimivw 3165	. . . . . . . 8 ⊢ (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥) → 𝑦 ∈ On)
9	8	abssi 4025	. . . . . . 7 ⊢ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥)} ⊆ On
10		limuni 6430	. . . . . . . . . . . 12 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴)
11	10	eqcomd 2772	. . . . . . . . . . 11 ⊢ (Lim 𝐴 → ∪ 𝐴 = 𝐴)
12		fveq2 6888	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐴 → (card‘𝑥) = (card‘𝐴))
13	12	eqcomd 2772	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐴 → (card‘𝐴) = (card‘𝑥))
14	13	biantrud 541	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐴 → (∪ 𝐴 = 𝐴 ↔ (∪ 𝐴 = 𝐴 ∧ (card‘𝐴) = (card‘𝑥))))
15		unieq 4888	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴)
16	15	eqeq1d 2768	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐴 → (∪ 𝑥 = 𝐴 ↔ ∪ 𝐴 = 𝐴))
17	1	pwid 4590	. . . . . . . . . . . . . . . . 17 ⊢ 𝐴 ∈ 𝒫 𝐴
18		eleq1 2854	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝒫 𝐴 ↔ 𝐴 ∈ 𝒫 𝐴))
19	17, 18	mpbiri 261	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝐴 → 𝑥 ∈ 𝒫 𝐴)
20	19	biantrurd 542	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐴 → (∪ 𝑥 = 𝐴 ↔ (𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴)))
21	16, 20	bitr3d 284	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐴 → (∪ 𝐴 = 𝐴 ↔ (𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴)))
22	21	anbi1d 643	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐴 → ((∪ 𝐴 = 𝐴 ∧ (card‘𝐴) = (card‘𝑥)) ↔ ((𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴) ∧ (card‘𝐴) = (card‘𝑥))))
23	14, 22	bitr2d 283	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → (((𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴) ∧ (card‘𝐴) = (card‘𝑥)) ↔ ∪ 𝐴 = 𝐴))
24	1, 23	spcev 3568	. . . . . . . . . . 11 ⊢ (∪ 𝐴 = 𝐴 → ∃𝑥((𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴) ∧ (card‘𝐴) = (card‘𝑥)))
25	11, 24	syl 18	. . . . . . . . . 10 ⊢ (Lim 𝐴 → ∃𝑥((𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴) ∧ (card‘𝐴) = (card‘𝑥)))
26		df-rex 3093	. . . . . . . . . . 11 ⊢ (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝐴) = (card‘𝑥) ↔ ∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ (card‘𝐴) = (card‘𝑥)))
27		rabid 3440	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ↔ (𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴))
28	27	anbi1i 636	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ (card‘𝐴) = (card‘𝑥)) ↔ ((𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴) ∧ (card‘𝐴) = (card‘𝑥)))
29	28	exbii 1881	. . . . . . . . . . 11 ⊢ (∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ (card‘𝐴) = (card‘𝑥)) ↔ ∃𝑥((𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴) ∧ (card‘𝐴) = (card‘𝑥)))
30	26, 29	bitri 278	. . . . . . . . . 10 ⊢ (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝐴) = (card‘𝑥) ↔ ∃𝑥((𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴) ∧ (card‘𝐴) = (card‘𝑥)))
31	25, 30	sylibr 237	. . . . . . . . 9 ⊢ (Lim 𝐴 → ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝐴) = (card‘𝑥))
32		fvex 6901	. . . . . . . . . 10 ⊢ (card‘𝐴) ∈ V
33		eqeq1 2770	. . . . . . . . . . 11 ⊢ (𝑦 = (card‘𝐴) → (𝑦 = (card‘𝑥) ↔ (card‘𝐴) = (card‘𝑥)))
34	33	rexbidv 3192	. . . . . . . . . 10 ⊢ (𝑦 = (card‘𝐴) → (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥) ↔ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝐴) = (card‘𝑥)))
35	32, 34	spcev 3568	. . . . . . . . 9 ⊢ (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝐴) = (card‘𝑥) → ∃𝑦∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥))
36	31, 35	syl 18	. . . . . . . 8 ⊢ (Lim 𝐴 → ∃𝑦∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥))
37		abn0 4344	. . . . . . . 8 ⊢ ({𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥)} ≠ ∅ ↔ ∃𝑦∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥))
38	36, 37	sylibr 237	. . . . . . 7 ⊢ (Lim 𝐴 → {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥)} ≠ ∅)
39		onint 7798	. . . . . . 7 ⊢ (({𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥)} ⊆ On ∧ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥)} ≠ ∅) → ∩ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥)} ∈ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥)})
40	9, 38, 39	sylancr 599	. . . . . 6 ⊢ (Lim 𝐴 → ∩ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥)} ∈ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥)})
41	4, 40	eqeltrid 2870	. . . . 5 ⊢ (Lim 𝐴 → ∩ 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝑥) ∈ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥)})
42	2, 41	eqeltrd 2866	. . . 4 ⊢ (Lim 𝐴 → (cf‘𝐴) ∈ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥)})
43		fvex 6901	. . . . 5 ⊢ (cf‘𝐴) ∈ V
44		eqeq1 2770	. . . . . 6 ⊢ (𝑦 = (cf‘𝐴) → (𝑦 = (card‘𝑥) ↔ (cf‘𝐴) = (card‘𝑥)))
45	44	rexbidv 3192	. . . . 5 ⊢ (𝑦 = (cf‘𝐴) → (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥) ↔ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (cf‘𝐴) = (card‘𝑥)))
46	43, 45	elab 3641	. . . 4 ⊢ ((cf‘𝐴) ∈ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥)} ↔ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (cf‘𝐴) = (card‘𝑥))
47	42, 46	sylib 221	. . 3 ⊢ (Lim 𝐴 → ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (cf‘𝐴) = (card‘𝑥))
48		df-rex 3093	. . 3 ⊢ (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (cf‘𝐴) = (card‘𝑥) ↔ ∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥)))
49	47, 48	sylib 221	. 2 ⊢ (Lim 𝐴 → ∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥)))
50		simprl 783	. . . . . . . 8 ⊢ ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴})
51	50, 27	sylib 221	. . . . . . 7 ⊢ ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → (𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴))
52	51	simpld 500	. . . . . 6 ⊢ ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → 𝑥 ∈ 𝒫 𝐴)
53	52	elpwid 4576	. . . . 5 ⊢ ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → 𝑥 ⊆ 𝐴)
54		simpl 488	. . . . . . 7 ⊢ ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → Lim 𝐴)
55		vex 3462	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
56		limord 6429	. . . . . . . . . . . 12 ⊢ (Lim 𝐴 → Ord 𝐴)
57		ordsson 7791	. . . . . . . . . . . 12 ⊢ (Ord 𝐴 → 𝐴 ⊆ On)
58	56, 57	syl 18	. . . . . . . . . . 11 ⊢ (Lim 𝐴 → 𝐴 ⊆ On)
59		sstr 3948	. . . . . . . . . . 11 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝐴 ⊆ On) → 𝑥 ⊆ On)
60	58, 59	sylan2 605	. . . . . . . . . 10 ⊢ ((𝑥 ⊆ 𝐴 ∧ Lim 𝐴) → 𝑥 ⊆ On)
61		onssnum 10043	. . . . . . . . . 10 ⊢ ((𝑥 ∈ V ∧ 𝑥 ⊆ On) → 𝑥 ∈ dom card)
62	55, 60, 61	sylancr 599	. . . . . . . . 9 ⊢ ((𝑥 ⊆ 𝐴 ∧ Lim 𝐴) → 𝑥 ∈ dom card)
63		cardid2 9958	. . . . . . . . 9 ⊢ (𝑥 ∈ dom card → (card‘𝑥) ≈ 𝑥)
64	62, 63	syl 18	. . . . . . . 8 ⊢ ((𝑥 ⊆ 𝐴 ∧ Lim 𝐴) → (card‘𝑥) ≈ 𝑥)
65	64	ensymd 9011	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝐴 ∧ Lim 𝐴) → 𝑥 ≈ (card‘𝑥))
66	53, 54, 65	syl2anc 596	. . . . . 6 ⊢ ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → 𝑥 ≈ (card‘𝑥))
67		simprr 785	. . . . . 6 ⊢ ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → (cf‘𝐴) = (card‘𝑥))
68	66, 67	breqtrrd 5144	. . . . 5 ⊢ ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → 𝑥 ≈ (cf‘𝐴))
69	51	simprd 501	. . . . 5 ⊢ ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → ∪ 𝑥 = 𝐴)
70	53, 68, 69	3jca 1146	. . . 4 ⊢ ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ (cf‘𝐴) ∧ ∪ 𝑥 = 𝐴))
71	70	ex 418	. . 3 ⊢ (Lim 𝐴 → ((𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥)) → (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ (cf‘𝐴) ∧ ∪ 𝑥 = 𝐴)))
72	71	eximdv 1950	. 2 ⊢ (Lim 𝐴 → (∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥)) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ (cf‘𝐴) ∧ ∪ 𝑥 = 𝐴)))
73	49, 72	mpd 16	1 ⊢ (Lim 𝐴 → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ (cf‘𝐴) ∧ ∪ 𝑥 = 𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2146  {cab 2744   ≠ wne 2961  ∃wrex 3092  {crab 3419  Vcvv 3458   ⊆ wss 3908  ∅c0 4289  𝒫 cpw 4567  ∪ cuni 4877  ∩ cint 4917  ∩ ciin 4962   class class class wbr 5114  dom cdm 5666  Ord word 6366  Oncon0 6367  Lim wlim 6368  ‘cfv 6543   ≈ cen 8949  cardccrd 9940  cfccf 9942

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-int 4918  df-iun 4963  df-iin 4964  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-se 5620  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7380  df-ov 7426  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-er 8703  df-en 8953  df-dom 8954  df-card 9944  df-cf 9946

This theorem is used by: (None)